Rumor Spreading and Conductance Flavio Chierichetti Silvio Lattanzi - - PowerPoint PPT Presentation

Rumor Spreading and Conductance

Flavio Chierichetti Silvio Lattanzi Alessandro Panconesi

Sapienza University of Rome

Why is rumor spreading fast in social networks?

• How to answer this question?
• How to define rumor spreading?
• What are social networks?

Rumor spreading

Rumor spreading

• Push, Pull and Push-Pull

Introduced in the contest of distributed database.

Demers, Greene, Hauser, Irish, Larson, Shenker, Sturgis, Swinehart, Terry, PODC 1987

• Basic mechanisms for information

dissemination in networks.

Push

Push

In each round, each informed node “pushes” the information to a neighbor chosen UAR.

Push

In each round, each informed node “pushes” the information to a neighbor chosen UAR.

Push

In each round, each informed node “pushes” the information to a neighbor chosen UAR.

Push

In each round, each informed node “pushes” the information to a neighbor chosen UAR.

Push

In each round, each informed node “pushes” the information to a neighbor chosen UAR.

Push

In each round, each informed node “pushes” the information to a neighbor chosen UAR.

Push

In each round, each informed node “pushes” the information to a neighbor chosen UAR.

Push

Pull

In each round, each uninformed node tries to “pull” the information from a neighbor chosen UAR.

Pull

In each round, each uninformed node tries to “pull” the information from a neighbor chosen UAR.

Pull

In each round, each uninformed node tries to “pull” the information from a neighbor chosen UAR.

Pull

In each round, each uninformed node tries to “pull” the information from a neighbor chosen UAR.

Pull

Push-Pull

Each node performs both actions on neighbor chosen UAR.

Push-Pull

Each node performs both actions on neighbor chosen UAR.

Push-Pull

Push-Pull

Each node performs both actions on neighbor chosen UAR.

Each node performs both actions on neighbor chosen UAR.

Push-Pull

Each node performs both actions on neighbor chosen UAR.

Push-Pull

Push Pull

• What are the completion times TPUSH, TPULL,

TPUSH-PULL?

• How many rounds will it take for each node to

know the information with probability 1 – o(1), assuming a worst-case source?

Performance

• TPUSH, TPULL = Θ(log n) if G = Kn

Frieze, Grimmet, Algorithms 1985

Previous work

• TPUSH, TPULL = Θ(log n) if G = Kn

Frieze, Grimmet, Algorithms 1985

• TPUSH ≤ O(n log n)

TPUSH ≤ O(Δ(G) ( diam(G) + log n )) TPUSH ≤ O(log n) in Hypercubes and G(n,p) Graphs

Feige, Peleg, Raghavan, Upfal, Algorithms 1990

Previous work

• TPUSH, TPULL = Θ(log n) if G = Kn

Frieze, Grimmet, Algorithms 1985

• TPUSH ≤ O(n log n)

TPUSH ≤ O(Δ(G) ( diam(G) + log n )) TPUSH ≤ O(log n) in Hypercubes and G(n,p) Graphs

Feige, Peleg, Raghavan, Upfal, Algorithms 1990

Previous work

Huge in Social Networks!

• TPUSH O(log n) in “quasi-regular” expanders

Berenbrink, Elsässer, Friedetzky, PODC 2008 Doerr, Friedrich, Sauerwald, ICALP 2009 Sauerwald, ISAAC 2007

• TPUSH-PULL O(log2n) in PA graphs

TPUSH-PULL O(poly(Φ-1 log n)) if conductance = Φ

Chierichetti, Lattanzi, Panconesi, ICALP 2009, SODA 2010

• Non uniform rumor spreading and conductance

Boyd, Ghosh, Prabhakar, Shah, IEEE Transaction on Information Theory 2006 Mosk-Aoyama, Shah, IEEE Transaction on Information Theory 2008

Previous work

• TPUSH O(log n) in “quasi-regular” expanders

Berenbrink, Elsässer, Friedetzky, PODC 2008 Doerr, Friedrich, Sauerwald, ICALP 2009 Sauerwald, ISAAC 2007

• TPUSH-PULL O(log2n) in PA graphs

TPUSH-PULL O(poly(Φ-1 log n)) if conductance = Φ

Chierichetti, Lattanzi, Panconesi, ICALP 2009, SODA 2010

• Non uniform rumor spreading and conductance

Boyd, Ghosh, Prabhakar, Shah, IEEE Transaction on Information Theory 2006 Mosk-Aoyama, Shah, IEEE Transaction on Information Theory 2008

Previous work

Social Networks are highly irregular!

• TPUSH O(log n) in “quasi-regular” expanders

Berenbrink, Elsässer, Friedetzky, PODC 2008 Doerr, Friedrich, Sauerwald, ICALP 2009 Sauerwald, ISAAC 2007

• TPUSH-PULL O(log2n) in PA graphs

TPUSH-PULL O(poly(Φ-1 log n)) if conductance = Φ

Chierichetti, Lattanzi, Panconesi, ICALP 2009, SODA 2010

• Non uniform rumor spreading and conductance

Boyd, Ghosh, Prabhakar, Shah, IEEE Transaction on Information Theory 2006 Mosk-Aoyama, Shah, IEEE Transaction on Information Theory 2008

Previous work

Connections to Spielman-Teng sparsification theory

Social networks

Empirical evidence

• Leskovec, Lang, Dasgupta and Mahoney give

empirical evidence that social networks have conductance

• Can we relate the performance of rumor

spreading algorithms with the conductance of the graph?

Ω

• 1

log n

Conductance

S C(S,V-S)

φ(G) = min

S⊆V vol(S)≤|E|

|C(S, V − S)| min(vol(S), vol(V − S))

Social networks high conductance

Social networks high conductance

Is rumor spreading fast on high conductance graphs?

TPUSH

Is the PUSH strategy fast?

TPUSH

Is the PUSH strategy fast?

TPUSH

Is the PUSH strategy fast? The star has constant conductance.

TPUSH

Is the PUSH strategy fast? The star has constant conductance.

TPUSH

Is the PUSH strategy fast? The star has constant conductance.

TPUSH

Is the PUSH strategy fast? The star has constant conductance.

TPUSH

Is the PUSH strategy fast? Coupon Collector!

TPUSH

Is the PUSH strategy fast? NO Coupon Collector!

TPULL

Is the PULL strategy fast?

TPULL

Is the PULL strategy fast?

TPULL

Is the PULL strategy fast? The central node has to PULL the information from the right node.

TPULL

The central node has to PULL the information from the right node. Is the PULL strategy fast? NO

TPUSH and TPULL in social networks

Social networks do not look like a star...

TPUSH and TPULL in social networks

Social networks do not look like a star... ... but there are low degree nodes connected only to high degree nodes.

TPUSH and TPULL in social networks

Social networks do not look like a star... ... but there are low degree nodes connected only to high degree nodes. SAME ISSUES!!!

TPUSH-PULL?

Theorem

Let G be a graph with conductance Φ, then w.h.p.

Theorem

Let G be a graph with conductance Φ, then w.h.p.

Lower bound

• Take any 3-regular graph of constant vertex

expansion of order Θ(nΦ) and diameter Θ(log n)

Lower bound

• Take any 3-regular graph of constant vertex

expansion of order Θ(nΦ) and diameter Θ(log n)

• Replace each edge with a path of length Θ(Φ-1)

Lower bound

• Take any 3-regular graph of constant vertex

expansion of order Θ(nΦ) and diameter Θ(log n)

• Replace each edge with a path of length Θ(Φ-1)
• The resulting graph will have order Θ(n), diameter

Θ(Φ-1 log n) and conductance Θ(Φ).

Upper bound

Let G be a graph with conductance Φ, then w.h.p.

Proof strategy

Key lemma

S

Proof strategy

S

We consider the process for O(Φ-1) steps. Key lemma

Proof strategy

S’

After Θ(Φ-1) steps with Θ(1) probability, if S’

is the new set of informed nodes, then Vol(S’) ≥ (1+ Ω(Φ)) Vol(S)

Key lemma

S

Proof strategy

S

We consider macro-phases composed by O(Φ-1)

steps

Proof strategy

We say that a macro-phase is successful if the volume increases by a factor of (1+ Φ).

S

We consider macro-phases composed by O(Φ-1)

steps

Proof strategy

After 1 successful macro-phase, we have:

Vol(S’) ≥ (1+ Ω(Φ)) Vol(S) S’

We consider macro-phases composed by O(Φ-1)

steps

Proof strategy

After 2 successful macro-phases, we have:

Vol(S’’) ≥ (1+ Ω(Φ))2 Vol(S) S’’

We consider macro-phases composed by O(Φ-1)

steps

Proof strategy

After Θ(Φ-1 log n) successful macro-phases, we have:

Vol(INFORMED) > Vol(G)/2

INFORMED

We consider macro-phases composed by O(Φ-1)

steps

Proof strategy

• A macro-phase is successful with constant

probability.

Proof strategy

• A macro-phase is successful with constant

probability.

• After O(Φ-1 log n) successful macro-phases, we

have Vol(INFORMED) ≥ Vol(G)/2

Proof strategy

• A macro-phase is successful with constant

probability.

• After O(Φ-1 log n) successful macro-phases, we

have Vol(INFORMED) ≥ Vol(G)/2

• Using the Chernoff bound after O(Φ-1 log n)

macro-phases, we have O(Φ-1 log n) successful macro-phases.

Proof strategy

After O(Φ-2 log n) steps we have Vol(INFORMED) > Vol(G)/2 w.h.p.

Proof strategy

After O(Φ-2 log n) steps we have Vol(INFORMED) > Vol(G)/2 w.h.p. E1

Proof strategy

E1 After O(Φ-2 log n) steps we have Vol(INFORMED) > Vol(G)/2 w.h.p.

Proof strategy

E1 E2 After O(Φ-2 log n) steps we have Vol(INFORMED) > Vol(G)/2 w.h.p.

Proof strategy

E1 E2 After O(Φ-2 log n) steps we have Vol(INFORMED) > Vol(G)/2 w.h.p.

Proof strategy

E1 E2 P(E1) = P(E2) After O(Φ-2 log n) steps we have Vol(INFORMED) > Vol(G)/2 w.h.p.

Proof strategy

• After O(Φ-2 log n) steps we have

Vol(INFORMED) > Vol(G)/2 w.h.p.

• After O(Φ-2 log n) steps each node pulls the

information from a set of nodes of Vol(G)/2

w.h.p.

• After O(Φ-2 log n) steps all the nodes have

the info w.h.p.

Key lemma

S’

After O(Φ-1) steps with constant probability, we

have that for the new set of informed nodes S’ Vol(S’) ≥ (1+ Ω(Φ)) Vol(S) S

Sketch of the proof

Idea: analyze what happens to each node in S in a macro-phase

S

Sketch of the proof

S

v0

Sketch of the proof

S

v0

Sketch of the proof

S

v0 G(v0)

Sketch of the proof

S

v1 G(v0) v0

Sketch of the proof

S

v1 G(v0) v0

Sketch of the proof

S

v1 G(v0) v0

Sketch of the proof

S

v1 G(v0) v0

Sketch of the proof

S

v1 G(v0) v0 G(v1)

Sketch of the proof

S

We define the following sets:

Sketch of the proof

We define the following sets:

• A⊆S, informed nodes that we still have to consider

S A

Sketch of the proof

We define the following sets:

• A⊆S, informed nodes that we still have to consider

S A

Sketch of the proof

We define the following sets:

• A⊆S, informed nodes that we still have to consider
• B⊇S, informed nodes at the current phase

S A B

Sketch of the proof

We define the following sets:

• A⊆S, informed nodes that we still have to consider
• B⊇S, informed nodes at the current phase

S A B

Sketch of the proof

We define the following sets:

• A⊆S, informed nodes that we still have to consider
• B⊇S, informed nodes at the current phase

S A B

Sketch of the proof

We define the following sets:

• A⊆S, informed nodes that we still have to consider
• B⊇S, informed nodes at the current phase

S A B

Sketch of the proof

S A B U

We define the following sets:

• A⊆S, informed nodes that we still have to consider
• B⊇S, informed nodes at the current phase
• U useful nodes in A

Sketch of the proof

S A B U

We define the following sets:

• A⊆S, informed nodes that we still have to consider
• B⊇S, informed nodes at the current phase
• U useful nodes in A

Sketch of the proof

The set of useful nodes U is

S A B U

U = UB(A) =

• v ∈ A | deg+

B(v)

deg(v) ≥ φ 2

Sketch of the proof

The set of useful nodes U is

S A B U

• 1. The cut (U, V - B) is a large part of the cut

(S, V - S), which has size at least Φ Vol(S).

U = UB(A) =

• v ∈ A | deg+

B(v)

deg(v) ≥ φ 2

Sketch of the proof

The set of useful nodes U is

S A B U

• 1. The cut (U, V - B) is a large part of the cut

(S, V - S), which has size at least Φ Vol(S).

• 2. And, furthermore, each node in U will have

constant probability of gaining a constant fraction of its edges in the cut.

U = UB(A) =

• v ∈ A | deg+

B(v)

deg(v) ≥ φ 2

Sketch of the proof

In order to get the key lemma we prove that for every macro- phase, and every v in U Pr

• G(v) ≥ 1

20 · deg+

B(v)

• ≥ 1 − e−1

U

Sketch of the proof

v

Bi

d+(v)

U

Sketch of the proof

v

Bi H

d+(v)

L

U

Sketch of the proof

v

By applying Chebyshev inequality with some arithmetic manipulation, we get that, in the PULL regime:

Bi

Pr

• g(v) > 1

20 · d+(v)

• ≥ 1

10

L

≥ 1/2 · d+(v)

U

Sketch of the proof

v

Bi

In the PUSH regime, we had:

Pr

• g(v) > 1

20 · d+(v)

• ≥ φ

10

H

≥ 1/2 · d+(v)

U

Sketch of the proof

v

Bi

So, in general,

Pr

• g(v) > 1

20 · d+(v)

• ≥ φ

10

d+(v)

Sketch of the proof

Since we go on for Φ-1 steps, Pr

• G(v) ≥ 1

20 · deg+

B(v)

• ≥ 1 − e−1

Upper bound

Let G be a graph with conductance Φ, then w.h.p.

The tighter bound

Let G be a graph with conductance Φ, then w.h.p.

Can the key lemma be improved?

After Θ(Φ-1) steps with Θ(1) probability, if S’

is the new set of informed nodes, then Vol(S’) ≥ (1+ Ω(Φ)) Vol(S)

Can the key lemma be improved?

After Θ(Φ-1) steps with Θ(1) probability, if S’

is the new set of informed nodes, then Vol(S’) ≥ (1+ Ω(Φ)) Vol(S)

Can the key lemma be improved?

After Θ(Φ-1) steps with Θ(1) probability, if S’

is the new set of informed nodes, then Vol(S’) ≥ (1+ Ω(Φ)) Vol(S)

Can the key lemma be improved?

After Θ(Φ-1) steps with Θ(1) probability, if S’

is the new set of informed nodes, then Vol(S’) ≥ (1+ Ω(Φ)) Vol(S)

∀v deg(v) ≃ Φ−1

Can the key lemma be improved?

After Θ(Φ-1) steps with Θ(1) probability, if S’

is the new set of informed nodes, then Vol(S’) ≥ (1+ Ω(Φ)) Vol(S)

deg = 2 deg ≃ Φ−1

Can the key lemma be improved?

After Θ(Φ-1) steps with Θ(1) probability, if S’

is the new set of informed nodes, then Vol(S’) ≥ (1+ Ω(Φ)) Vol(S)

No?

Stronger key lemma

S

For some p ≥ Φ, after O(1/p) steps with constant probability,

we have that for the new set of informed nodes S’

vol(S′) ≥

• 1 + Ω
• φ

p log2 φ−1

• · vol(S)

Conclusion

• We studied the rumor spreading problem in graph
• f conductance Φ.
• We showed that the PUSH and the PULL strategies are

not fast,

• and that the PUSH-PULL strategy is fast, and we gave an

almost tight bound for its performance.

Conclusion

• We studied the rumor spreading problem in graph
• f conductance Φ.
• We showed that the PUSH and the PULL strategies are

not fast, (fast if some kind of “degree uniformity” exists)

• and that the PUSH-PULL strategy is fast, and we gave an

almost tight bound for its performance.

Open problems

• Find a tight bound for the PUSH-PULL strategy.
• Study the relationship between rumor spreading and

vertex expansion.

• Can the PUSH strategy inform efficiently a large part of

a social network?

Thank you! Questions?